Lower bounds for time-dependent layout measurements

We prove lower bounds on the accounting exponent of OpenAI's layout interface (from The Exponential Time Hypothesis is False) for bounded-incidence, balanced-cut expanding scopes. The proof couples arbitrary time-dependent measurement spaces through a determinant. Under a capacity condition the charged exponent is at least a constant times (log log n)3/2/√log n. We also prove a small-measurement bound in auxiliary coordinates and a separate partner-time inequality. These are not general algorithmic running-time lower bounds. The record contains the paper (PDF and LaTeX source) and exact verification scripts.